- #1
iRaid
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Homework Statement
Find the following Fourier series in trigonometric form.
Homework Equations
$$y(t)=a_0+\sum\limits_{n=1}^{\infty} a_n cos(n\omega_{0}t)+b_n sin(n\omega_{0}t)$$
The Attempt at a Solution
The graph above is represented by the function:
$$
x(t) = \left\{
\begin{array}{ll}
-t+1 & \quad 0 < t \leq 1 \\
0 & \quad 1 < t \leq 2
\end{array}
\right.
$$
##T_0=2## and ##\omega_{0}=\pi##
To find ##a_0##:
$$a_0=\frac{1}{T_{0}}\int_{0}^{T_{0}} x(t)dt = \frac{1}{2}\int_{0}^{1} (-t+1)dt = \frac{1}{4}$$
The integral of 0 can be ignored since it equals 0.
To find ##a_n##:
$$a_n=\frac{2}{T_{0}}\int_{0}^{T_0}x(t) cos(n\omega_{0}t)dt=\frac{2}{2}\int_{0}^{1} (-t+1)cos(\pi nt)dt=\int_{0}^{1} -tcos(\pi nt)dt +cos(\pi nt)dt$$
To find ##b_n##
$$b_n=\frac{2}{T_{0}}\int_{0}^{T_{0}} x(t)sin(n\omega_{0}t)dt = \frac{2}{2}\int_{0}^{1} (-t+1)sin(\pi nt)dt=\int_{0}^{1} -tsin(\pi nt)+sin(\pi nt)dt$$
I am assuming that the ##T_0## in these equations are for the period of each function, since the function x(t)=0 is from 1 to 2, I only use 1 for ##T_0## for the function x(t)=-t+1.
Now I can find these integrals for ##a_n## and ##b_n## using integration by parts, but I'm wondering if I am going along with this correctly and what I will do once I find it (the summation really confuses me).Thank you for any help!
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